# Natural Deduction

Use natural deduction to prove that each of the following arguments is valid. Each of the proofs in this section require between 2-5 steps. (2 points each)

(1) 1. A

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2. pA _ Bq ùñ C

∴ 3. C

(2) 1. A ^ B

2. C ^ D

∴ 3. A ^ D

(3) 1. A ùñ B

2. B

3. A ùñ C

∴ 4. C

(4) 1. A _ B

2. A ùñ C

3. B ùñ pD ^ Eq

∴ 4. C _pD ^ Eq

(5) 1. A ^ B

2. C _ D

3. C ùñ A

∴ 4. D

(6) 1. A ùñ pB _ Cq

2. pB _ Cq ùñ D

3. D

∴ 4. A

(7.) 1. A

2. pB _ Aq ùñ C

3. C ^ D

∴ 4. B

(8) 1. A ùñ B

2. B

∴ 3. A ^ B

(9.) 1. A ^ B

2. B ùñ C

3. C ùñ pD ^ Eq

∴ 4. E

(10) 1. D

2. C

3. D ùñ pA _ Cq

∴ 4. A

1

Rectangle

2 Direct Proofs 2

Use natural deduction to prove that each of the following arguments is valid. These proofs are more complex and require more steps. (6 points each)

(11.) 1. pB _ Aq ùñ pD ^ Eq

2. C

3. pC _ Fq ùñ pA ^ Bq

∴ 4. E

(12.) 1. A ùñ B

2. B _ C

3. D ^ B

4. pC ^ Aq ùñ E

∴ 5. pD ^ Eq^ C

(13.) 1. A ðñ B

2. B _ D

3. pA _ Eq ùñ C

4. pD ^ Eq^ F

∴ 5. C

14. 1. A

2. pA _ Cq ùñ B

3. C

4. D ùñ B

∴ 5. D ^ A

(15.) 1. A ùñ pB ^pC _ Dqq

2. pE _ Aq ùñ pD ùñ Fq

3. E ùñ pG _ Hq

4. pD ùñ Fq

∴ 5. p pD ùñ Fqq^ppB ^pC _ Dqq_pG _ Hqq

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